l. e. j. brouwer - consciousness, philosophy, and mathematics

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L. E. ]. BROUWER (Blaricum)

CONSCIOUSNESS, PHILOSOPHY, AND MATHEMATICS

First of all an account should be rendered of the phases consciousness has to pass through in its transition from its deepest home to the exterior world in which we cooperate and seek mutual understanding. This account does not imply mutual understanding and in some way may remain a soliloquy. The same can be said of some other parts of this lecture too.

Consciousness in its deepest home seems to oscillate slowly, will-lessly, <1.nd reversibly between stillness and sensation. And it seems that only the status of sensation allows the initial phenomenon of the said transition. This initial phenomenon is a move of time. By a move of time a present sensation gives way to another present sensation in such a way that consciousness retains the former one as a past sensation, and moreover, through this distinction between present and past, recedes from both and from stillness, and becomes mind.

As mind it takes the function of a subject experiencing the present as well as the past sensation as object. And by reiteration of this two-ityphenomenon, the object can extend to a world of sensations of motley plurality.

In measure of the irreversibility with which the subject has receded from an element of the object, this element loses its egoicity, i.e. gets estranged from the subject, and in measur-e of this estrangement, mind becomes disposed to desire and apprehension, and consequently to positive or negative conative activity with respect to the element in question.

In the world of sensation experienced by mind, the free-will-phenomenon of causal attention occurs. It performs identifications of different sensations and of different complexes of sensations, and in this way, in a dawning atmosphere of forethought, creates iterative complexes of semations. An iterative complex of sensations, whose elements have an invariable order of succession in time, whilst if one of its elements occurs, all following elements are expected to occur likewise, in the right order of succession, is called a causal sequence.

On the other hand there are iterative complexes of sensations whose elements are permutable in point of time. Some of them are completely estranged from the subject. They are called things. For instance individuals, i .e. human bodies, the home body of the subject included, are things. Things may be, or may not be, indissolubly connected w ith egoic sensations. The whole of egoic sensations indissolubly connected with an individual, ts called the soul of the corresponding human being. The soul connected

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with the subject-individual is rather latent, but manifest in sensations of vocation and of inspiration.

The whole of things is called the exterior world of the subject. Causal sequences, each of whose elements contains a thing, are called exterior causal sequences.

Causal attention allows the development of the conative act1v1ty of the subject from spontaneous effort to forethinking enterprise by means of the free-will-phenomenon of cunning act. The cunning act consists in this, that in a causal seequence of eventualities, a later element not conatively attainable in a spontaneous way but nevertheless desired (the aim), is realized indirectly by bringing about an in itself perhaps non-desirable but conatively attainable earlier element of the sequence (the means), and in its wake obtaining the desired element as its consequence. A causal sequence employed in this way is called a useful causal sequence. As a matter of course aim as well as means may be of a negative (averting) character. Mood and temper directing cunning acts are essentially different from spontaneous desire and apprehension.

Since (positive and negative) conative activity is mainly directed towards things, individuals included, the cunning act chiefly operates with exterior causal sequences.

By means of its cunning acts, the subject creates a causal sphere of influence which on the one hand it protects by an activity of destroying things endangering useful causal sequences, and which on the other hand it extends by an activity of constructing things capable of new useful causal sequences.

As a matter of course sources of disappointment with cunning acts are numerous. In the first place direct fulfilment of (positive or negative) desire through spontaneous activity is never equalled by its appeasement in a circuitous way. Furthermore causal attention meets with a certain resistance from the part of the object, so that over and over aga;in confidence in causal sequences meets with unexpected and inexplicable deceptions, notwithstanding all effort at protection. Moreover all causal sequences are affected with .inaccuracies, so that a concatenation of causal sequences need not necessarily constitute another causal sequence.

However, in spite of these disappointments, mind, once having taken to causal attention, remains in a lasting causal tension, impelling alternately to causal thinking, i.e. attention toward discovering causal sequences and possibilities of creating or protecting causal sequences, and to causal acting, i .e. acting, generally cunning acting, in consequence of causal thinking.

In this connection there is a phenomenon of play, occurring when conative activity or causal thinking or acting is performed playfully, i.e. without inducement of either desire or apprehension or vocation or inspiration or compulsion.

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Causal attention repeatedly leads to identification of sensation complexes originating from causal acts of the subject, with sensation complexes experienced in causal connection with other individuals. On account of this identification the latter sensation complexes are called acts of those other individuals. Moreover causal acts of the subject and such of numerous other individuals influence each other in a high degree; many causal acts of many individuals even seem only to hav,e possibility and sense as items of organized cooperation of smaller or larger groups of individuals; the share of the subject in that cooperation seems to be of no other nature than that of the individuals of the object.

Systems of causal thinking underlying such cooperative causal acts, are far more complicated than those inducing individual causal acts. Prominent amongst the former is scientific thinking, which in an economical and efficient way catalogues extensive groups of cooperative causal sequences. And this scientific thinking, in particular when concerned with technique, is bas,ed on mathematics.

Mathematics comes into being, when the two-ity created by a move of time is divested of all quality by the subject, and when the remaining empty form of the common substratum of all two-ities, as basic intuition of mathematics, is left to an unlimited unfolding, creating new mathematical entities in the shape of predeterminately or more or less freely proceeding infinite sequences of mathematical entities previously acquired, and in the shape of mathematical species i.e. properties supposable for mathematical entities previously a:cquired and satisfying the condition that if they are realized for a certain mathematical entity, they are also realized for all mathematical entities which have been defined equal to it.

The significance of mathematics with regard to scientific thinking mainly consists in this that a group of observed causal sequences can oft.en be manipulated more easily by extending its of-quality-divested mathematical substratum to a hypothesis, i.e. a more comprehensive and more surveyable mathematical system. Causal sequences represented in abstraction in the hypothesis, but so far neither observed nor found observable, often find their realization later on.

The organization of a group of individuals into a cooperation consists in a wire-netting of will-transmission. At primitive stages of civilization and in primitive man-to-man relations this transmission of will from individual to individual is brought about by simple gestures or primitive animal sounds. But in more developed organization of the groups concerned the acts to be imposed become too much differentiated and too complicated to be indicated exclusively in such a simple way. In order to be able under these circumstances still to direct the trade by means of requesting or commanding (auditive or visual) signals, the totality of laws, decrees, objects and theories concerned with the acts enjoined upon the organized individuals, is subjected to a causal attention, the linguistic causal attention, and the

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elements of the mathematical system resulting from this causal attention, are indicated by linguistic basic signs. From these basic signs, by means of grammatical rules taken from the said mathematical system, the organized languages allowing the ·extremely differentiated and complicated willtransmissions required by civilization have been constructed. And these languages not only consolidate the wire-netting of will-transmission, but also suggest its continual extension. Of course, much of the stability and exactness which according to grammar and lexicon a language seems to possess is lost in practical life, because the totality of cooperations requires far more basic notions than language has to off er basic words and word connections. On the other hand stability and exactness of language is not necessary in practical life, because in every organization routine engenders a sort of collective will, making good understanders to whom a word suffices.

In the preceding, account has been rendered of three successive phases of the exodus of consciousness from its deepest home. Of these phases the naive one was opened with the creation of the world of sensations, the isolated causal one with the setting in of causal activity, and the social one with being involved in cooperation with other individuals. Regression from the third to the second phase appears to be frequent and easy, but from either of these regression to the naive phase seems hard to realize, more easily a temporary refluence to the d eepest home leaving aside naivety, through the free-will-phenomenon of detachment-concentration. The question arises, whether and where, on and after this exodus of consciousness, beauty, mutual understanding, wisdom and truth can be found.

In causal thinking and acting beauty will hardly be found. Things as such are not beautiful, nor is their ·domination by shrewdness. Therefore satisfaction at efficacy of causal acts or systems of causal acts or at discoveries of new causal sequences is no sensation of beauty.

But in the first phase of the exodus there is beauty in the joyful miracle of the self-revelation of consciousness, as apparent in egoic elements of the object found in forms and forces of nature, in particular in human figures and human destinies, human splendour and human misery.

And in the second and third phase there is beauty in remembrance of the miracle of bygone naivety, remembrance evoked either by reverie through a haz·e of wistfulness and nostalgia, or by (self-created or encountered) works of art, or by certain kinds of science. Such science evoking beauty reveals or playfully mathematizes naively perceptible forms and laws of nature, after having approached them with attentive reverence, and with a minimum of tools. And such science evoking beauty, through its very reverence, rejects expansion of human domination over nature.

Furthermore in the second and the third phase there is constructional

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beauty, which sometimes appears when the activity of constructing things is exerted playfully, and, thus getting a higher degree of freedom of unfolding, creates things evoking sensations of power, balance, harmony, and acquiescence with the exterior world.

But the fullest constructional beauty is the introspective beauty of mathematics, where instead of elements of playful causal acting, the basic intuition of mathematics is left to free unfolding. This unfolding is not bound to the exterior world, and thereby to finiteness and responsibility; consequently its introspective harmonies can attain any degree of richness and clearness.

In every cooperation in which acts of the subject are concerned, to causal attention it seems that in the system of cooperative causal acts concerned the share of the subject, considered as share of the subject individual, is of no other nature relative to things than that of the object individuals concerned in the cooperation. And this finds expression in the language of the cooperation concerned. Again, to causal attention it seems that also the non-cooperative actions of the subject, considered as actions of the subject individual, firstly are of no other nature relative to things than those of the object individuals concerned in the cooperation, and secondly neither very much differ in nature relative to things from those of a great deal of object individuals not concerned in the coopeeration. And this finds expressibn in the language of the cooperation concerned in such a way that the part assigned to the subject individual in this language is analogous to those assigned to object individuals, whereas the subject itself is ignored in it. In this way civilized languages, mostly being cooperativ·e languages, suggest a sameness for such totally different phenomena as acts of the subject and acts of object individuals are.

And this suggestion is intensified by the misleading terms civilized languages use to characterize the behaviour of individuals in general. It is not unreasonable to derive this behaviour from "reason". But unreasonable to derive it from "mind". For by the choice of this term the subject in its scientific thinking is induced to place in each individual a mind with freewill dependent on this individual, thus elevating itself to a mind of second order ·experiencing incognizable alien consciousnesses as sensations. Quod non est. And which moreover would have the consequence that the mind of second order would causally think about the pluralified mind of first order, then cooperatively study the science of the pluralified mind, and in consequence of this study assign a mind of second order with sensation of alien consciousnesses to other individuals, thus once more elevating itself, this time to a mind of third order. And so on. Usque ad infinitum. And this nonsense would still go further. In the group of individuals Ii , I 2, . . • In, besides the primary minds of first order M 1 , M 2, • . • Mn, being sensations of the subject mind of second order, for every k and l which are natural numbers < n, the mind Mk 1 occurs, 1.e. the sensation

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experienced by !k as a mind of second order from It . Likewise, besides the primary minds of second order Mkz, being sensations of the subject mind of third order, for every e, a and -i: which are natural numbers < n, the mind M (}ar occurs, i.e. the sensation experienced by I e as a mind of third order from Mar. And so on. Usque ad infinitum. Moreover, with respect to behaviour, the variation from individual to individual, only in degree, not in essence, differs from the variation from individual to animal, so that as a consequence of the plurality of mind, a mind would have to be assigned to animals as well.

Since there is no plurality of mind, so much the less is there a science of the plural mind. Only a psychology of man and animal, which as an extension of physiology, studies automatic living organisms, without mind and without free-will. To these the subject individual belongs as well, notwithstanding its special role as bearer of joy and pain, and of phenomena accompanying emotions, thoughts and acts of the subject. For in spite of its dominating position the subject has a domain of describability which, compared to that of the object, is a Citta del Vaticano.

In default of a plurality of mind, there is no exchange of thought either. Thoughts are inseparably bound up with the subject. So-called communicating-of-thoughts to somebody, means influencing his actions. Agreeing with somebody, means being contented with his cooperative acts or having entered into an alliance. Dispelling misunderstanding, means repairing the wire-netting of will-transmission of some cooperation. By so-called exchange of thought with another being the subject only touches the outer wall of an automaton. This can hardly be called mutual understanding. Only through the sensation of the other's soul sometimes a deeper approach is experienced. And when wisdom revealed by the beauty of this sensation, finds expression in the antiphony of words exchanged, then the·re may be mutual understanding.

Apart from the soul ev.ery expose on the sense and essence of life is a soliloquy, and every discussion about the pluralified mind is a game of dialectics in the arena of the collective hypothesis of a collective supersubject experiencing an objective world which exists independently of the supposed human subjects that appear and disappear in it, which remains when all supposed human subjects have vanished, and would be, even if there had never been human subjects called into existence.

Searching for wisdom, we may find it in knowing that ·Causal thinking and acting is non-beautiful and hard to justify, and that in the long run it brings disappointment. And in knowing that the exterior world with its innumerable individuals and with its hypertrophied cooperation is wedded to mind, its disharmonies reflecting mind's free-will-guilt.

As a consequence of this knowing the exterior world and one's own position in it are accepted as they are, so that towards the exterior world generally only acts as reversible as possible aiming at maintenance, but no

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acts let alone causal acts aiming at change, are undertaken of one's free will. Repair of disadjustments, averting of danger and relief of need, all this negative intervening in human society is justified in itself and sometimes prescribed. But positive activity to change the structure of human society gov·emed by so many unknown forces, will always be checked by the self-admonition: "not to improve her work has Providence placed thee in this world", and only vocation and inspiration tested in detachment-concentration will be stronger than this admonition.

For the actual expansion of the already hypertrophied world cooperation (which in its trivial final aims of mass-comfort and mass-security has not got much justification) responsibility will be declined. Therefore leading positions in this cooperation will not be aspired to.

Neither can responsibility be assumed for curtailment of freedom, one's own or other people's. Therefore one will only reluctantly join a clique or union, these generally impairing lirberty of action and spontaneity in conduct of life.

Power over fellow-creatures will be avoided. Firstly because one would get mixed up with limitations of other people's liberty of action. And secondly because those fellow-creatures are part of the reflex image held out to mind from its deepest home, therefore have to be respected, and must not be judged, let alone condemned, despised or rejected, even if they are enemies to be fought against.

Eastern devotion has perhaps better expressed this wisdom than any western man could have done. For instance in the following passages of the Bhagavad-Gita 1) which even in translation have conserved their electrifying power:

"A man should not hate any living creature. Let 'him be friendly and compassionate to all. He must free himself from the delusion of I and mine. He must accept pleasure and pain with an equal tranquillity. He must be forgiving, ever-contented, self-controlled, united constantly. His resolve must be unshakable."

"He neither molests his fellow-men, nor allows himself to become disturbed by the world. He is no longer swayed by joy and envy, anxiety aad fear."

"He is pure and independent of the body's desire. He is able to deal with the unexpected : prepared for everything, unperturbed by anything. He i5 neither vain nor anxious about the results of his actions."

"He does not desire to rejoice in what is pleasant. He does not dread w'hat is unpleasant, or grieve over it. He remains unmoved by good or evil fortune."

1) Quoted from the English version by Swami Prabhavananda and Christopher Isherwood, London, Phoenix House, 1947.

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"His attitude is the same toward friend and foe. He is indifferent to honour and insult, heat and cold, pleasure and pain. He is free from attachment. He values praise and blame equally. He can control his speech. He is content with whatever he gets. His home is everywhere and nowhere."

The categorical imperative prescribing the aforesaid attitude towards life has its counterpart in a sceptical prognosis that mankind, possessed by the delusion of causality, will slide away in a deteriorative process of overpopulation, industrialization, serfdom, and devastation of nature, and that when hereby first its spiritual and then its physiological conditions of life will have been destroyed, it will come to its end like a colony of bacteria in the earth crust having fulfilled its task.

All this though timeless art and perennial philosophy continually suggest that the unknown forces governing the destiny of individual and community, are not subject to causality; that in particular the ways of fate cannot be paved with causality, and that security is as unattainable as it is unworthy; that intensification of organization increases vulner�bility, that new vulnerability asks for protection through new organization, and that thus for organization which is believed in, there is no end of growth ; finally that if the delusion of causality could be thrown off, nature, gradually resuming her rights, would be (except for her bondage to destiny) generous and forgiving to a mankind decausaliz·ed and subsiding to more modest and more ha:rmonious proportions.

Of course art and philosophy continually illustrating such wisdom cannot participate in cooperation, and should not communicate with cooperation, in particular should not communicate with the state. Supported by the state, they will lose their independence and degenerate.

The recognition of a cooperative world captured in the delusion of causality as a reflex of mind's guilt, does not imply eternal bondage to that world. Consequently, the way along which the deepest home was left, seeming to be blocked for final return, there may be wisdom in a patient tending towards reversible liberation from participation in cooperative trade and from intercourse presupposing plurality of mind. It seems that this mere tendency favours evaporation of desire and fear, so that gradually non-cooperative activity is allayed, cooperative causal acts are automatized, the world of things faints away, the joy of beauty pales, egoic elements no longer bind attention, and the home body grows more and more frugalized. What remains of non-cooperative conativ·e activity, seizes every opportunity for a further disengaging from cooperative trade and further anachoresis of the home body. There is no hesitation to leave what is beloved, for neither beauty nor egoic alliance needs causal proximity. Though there is sadness when in a receding distance naivety vanishes for ever. But perhaps at the end of the journey the deepest home vaguely beckons.

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From the above report, especially from the rejection of the hypothesis cf plurality of mind, follows that truth is only in reality i.e. in the present and past ,experiences of consciousness. Amongst these are things, qualities of things, emotions, rules (state rules, cooperation rules, game rules) and deeds (material deeds, deeds of thought, mathematical deeds). But expected experiences, and experiences attributed to others are true only as anticipations and hypotheses; in their contents there is no truth.

Truths often are conveyed by words or word complexes, generally borrowed from cooperation languages, in such a way that for the subject together with a certain word or word complex always a definite truth is evoked, and that object individuals behave accordingly. Further there is a system of general rules called logic enabling the subject to deduce from systems of word complexes conveying truths, other word complexes generally conveying truths as well. Causal behaviour of the subject (isolated as well as cooperative) is affected by logic. And again object individuals behave accordingly. This does not mean that the additional wnrrl complexes in question convey truths before these truths have been experienced, nor that these truths alwaxs can be experienced. In dher words, logic is not a reliable instrument to discover truths and cannot deduce truths which would not be accessible in another way as well.

The above point of view that there are no non-experienced truths and that logic is not an absolutely reliable instrument to discov·er truths, has found acceptance with regard to mathematics much later than with regard to practical life and to science. Mathematics rigorously treated from this point of view, and deducing theorems exclusively by means of introspective construction, is called intuitionistic mathematics. In many respects it deviates from classical mathematics. In the first place because classical mathematics uses logic to generate theorems, believes in the existence of unknown truths, and in particular applies the principle of the excluded third expressing that every mathematical assertion (i.e. every assignment of a mathematical property to a mathematical entity) either is a truth or cannot be a truth. In the second place because classical mathematics confines itself to predeterminate infinite sequences for which from the beginning the nth element is fixed for each n. Owing to this confinement classical mathematics, to define real numbers, has only predeterminate convergent infinite sequences of rational numbers at its disposal. Out of real numbers defined in this way, only subspecies of "ever unfinished denumerable" species of real numbers can be composed by means of introspective construction. Such ever unfinished denumerable species all being of measure zero, classical mathematics, to create the continuum out of points, needs some logical process starting from one or more axioms. Consequently we may say that classical analysis, however appropriate it be for technique and science, has less mathematical truth than intuitionistic analysis performing the said composition of the continuum by considering the species of freely proceed-

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ing convergent infinite sequences of rational numbers, without having recourse to language or logic.

As a matter of course also the languages of the two mathematical schools diverge. And even in those mathematical theories which are covered by a neutral language, i.e. by a language understandable on both sides, either school operates with mathematical entities not recognized by the other one: there are intuitionist structures which cannot be fitted into any classical logical frame, and there are classical arguments not applying to any introspective image. Likewise, in the theories mentioned, mathematical entities recognized by both parties on each side are found satisfying theorems which for the other schoQl are either false, or senseless, or even in a way contradictory. In particular, theorems holding in intuitionism, but not in classical mathematics, of ten originate from the circumstance that for mathematical entities belonging to a certain species, the possession of a certain property imposes a special character on their way of development from the basic intuition, and that from this special character of their way of development from the basic intuition, properties ensue which for classical mathematics are false. A striking example is the intuitionist theorem tha,t a full function of the unity continuum, i.e. a function assigning a real number to every non-negative real number not exceeding unity, is necessarily uniformly continuous.

To elucidate the consequences of the rejection of the principle of the excluded third as an instrument to discover truths, we shall put the wording of this principle into the following slightly modified, intuitionistically more adequate form, called the simple principle of the excludeid third:

Every assignment -r of a property to a mathematical entity can be j u d g e d, i.e. either proved or reduced to absurdity.

Then for a single such assertion -r the enunciation of this principle is non-contradictory in intuitionistic as well as in classical mathematics. For, if it were contradictory, then the absurdity of -r would be true and absurd at the same time, which is impossible. Moreover, as can easily be proved, for a finite number of such assertions -r the simultaneous enunciation of the principle is non-contradictory likewise. However, for the simultaneous enunciation of the principle for all elements of an arbitrary species of such assertions -r this non-contradictority cannot be maintained.

E.g. from the supposition, for a definite real number ci, that the assertion: c1 is rational, has been proved to be either true or contradictory, no contradiction can be deduced. Furthermore, ci, c2, • • • cm being real numbers, neither the simultaneous supposition, for each of the values 1 , 2 , . . . m of v, that the assertion: c� is rational, has been proved to be either true or contradictory, can lead to a contradiction. However, the simultaneous supposition for all real numbers c that the assertion: c is rational, has been proved to be either true or contradictory, does lead to a contradiction.

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Consequently if we formulate the complete principle of the excluded third as follows :

If a, b and c are species of mathematical entities, if further both a and b form part of c, and if b consists of those elements of c which cannot belong to a, then c is identical with the union of a and b,

the latter principle is contradictory.

A corollary of the simple principle of the excluded third says that if for an assignment T of a property to a mathematical entity the non-contradictority, i.e. the absurdity of the absurdity, has been established, the truth of T can be demonstrated likewise.

The analogous corollary of the complete principle of the excluded third [3] is the principle of reciprocity of complementarity, running as follows:

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If a, b and c are species of mathematical entities, if further a and b form part of c, and if b consists of the elements of c which cannot belong to a, then a consists of the elements of c which cannot belong to b.

Another corollary of the simple principle of the excluded third is the simple principle of testability saying that every assignment T of a property to a mathematical entity can be t e s t e d, i.e. proved to be either noncontradictory or absurd.

The analogous corollary of the complete principle of the excluded third is the following complete principle of testability:

If a, b, d and c are species of mathematical entities, if each of the species a, b, and d forms part of c, if b consists of the elements of c which cannot belong to a, and d of the elements of c which cannot belong to b, then c is identical with the union of b and d.

For intuitionism the principle of the excluded third and its corollaries are assertions a about assertions r, and thes·e assertions a only then are "realized", i.e. only then convey truths, if these truths have been experienced.

Each assertion T of the possibility of a construction of bounded finite character in a finite mathematical system furnishes a case of realization of the principle of the excluded third. For every such construction can be attempted only in a finite number of particular ways, and each attempt proves successful or abortive in a finite number of steps.

If the assertion of an absurdity is called a negative assertion, then each negative assertion furnishes a case of realization of the principle of reciprocity of complementarity. For, let a be a negative assertion, indicating the absurdity of the assertion {3. As, on the one hand, the implication of the truth of an assertion a by the truth of an assertion b implies the implication of the absurdity of b by the absurdity of a, whilst, on the other hand, the truth of f3 implies the absurdity of the absurdity of {3, we conclude that the absurdity of the absurdity of the absurdity of {3, i.e. the non-contradictority of a, implies the absurdity of {3, i.e. implies a.

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In consequence of this realization of the principle of reciprocity of complementarity the principles of testability and of the excluded third are equivalent in the domain of negative assertions. For, if for a the principle of testability holds, this means that either the absurdity of the absurdity of f3 or the non-contradictority of the absurdity of {3, i.e., by the preceding paragraph, that either the absurdity of the absurdity of {3 or the absurdity of {3, i.e. either the absurdity of a or a can be proved, so that a satisfies the principle of the excluded third.

To give some examples refuting the principle of the excluded third and its corollaries, we introduce the notion of a drift. By a drift we understand the union y of a convergent fundamental sequence of real numbers c1 (r), c2 (Y), • . . , called the counting-numbers of the drift, and the limitingnumber c (r) of this sequence, called the kernel of the drift, all countingnumbers lying apart 1) from each other and from the kernel. If c"' (r) <0 c (r) for each v, the drift will be called left-winged. If c,,, (r) 0> c (Y) for each v, the drift will be called right-winged. If the fundamental sequence c1 (r), c2 (r), . . . is the union of a fundamental sequence of left counting-numbers l1 (y), /2 (y), . . . such that /,,, (r) <0 c (r) for each v, and a fundamental sequence of right counting-numbers di (r), d2 (r), . . . such that d,,, (r) 0> c (r) for each v, the drift will be called two-winged.

Let a be a mathematical assertion so far neither tested nor recognized as testable. Then in connection with this assertion a and with a drift r the creating subject can generate an infinitely proceeding sequence R(y, a) of real numbers c1 (Y, a) , c2 (l', a), . • . according to the following direction: As long as during the choice of the cn (Y, a) the creating subject has experienced neither the truth, nor the absurdity of a, each cn (r, a) is chosen equal to c(r). But as soon as between the choice of C r - I (y, a) and that of Cr (y, a) the creating subject has experienced either the truth or the absurdity of a, cr (r, a), and likewise C r + "' (r, a) for each natural number v, is chosen equal to c r (y). This sequence R(y, a) converges to a real number D(r, a) which will be called a direct checking-number of y through a.

Again, in connection with a and with a two-winged drift y the creating subject can generate an infinitely proceeding sequence S(r, a) of real numbers w1 (l', a), w2 (l', a), . . . . according to the following direction: As long as during the choice of the Wn (r, a) the creating subject has

1) If for two real numbers a and b defined by convergent infinite sequences of

rational numbers ai, a2, . • . and bi,, b2, . . . respectively, two such natural numbers m and n can be calculated that b,., - a,., > 2- n for v '.:: m, we write b o > a and a <o b, and a and b are said to lie apart from each other. If a = b is absurd, we write a =fa b. If a <o b is absurd, we write a '.:: b. If both a = b and a <o b are absurd, we write

a > b. The absurdities of a <o b and a < b prove to be mutually equivalent, and the

absurdity of a '.:: b proves to be equivalent to a < b.

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experienced neither the truth, nor the absurdity of a, each Wn (r, a) is chosen equal to c (r) . But as soon as between the choice of W r - 1 (r, a) and that of wr (r, a) the creating subject has experienced the truth of a, w, (r, a) , and likewise W r + 7 (r, a) for each natural number Y, is chosen equal to dr (r). And as soon as between the choice of W s - 1 (r, a) and that of W s (r, a) the creating subject has experienced the absurdity of a,

Ws (r, a) , and likewise W s + 7 (r, a) for each natural number Y, is chosen equal to ls (r). This sequence S(r, a) converges to a real number E(r, a) which will be called an oscillatory checking-number of r through a.

Let r be a right-winged drift whose counting-numbers are rational. Then the assertion of the rationality of D(r, a) is testable, but not judgeable, and its non-contradictority is not equivalent to its truth. Furhermore we have D(r, a) > c (r), but not D(r, a) 0> c (r).

Let r be a two-winged drift whose right counting-numbers are rational, and whose left counting-numbers are irrational. Then the assertion of the rationality of E(r, a) is neither judgeable, nor is it testable, nor is its noncontradictority equivalent to its truth. Furthermore E{r, a) is neither > c(y), nor < c(r).

The long belief in the universal validity of the principle of the excluded third in mathematics is considered by intuitionism as a phenomenon of history of civilization of the same kind as the old-time belief in the rationality of n or in the rotation of the firmament on an axis passing through the earth. And intuitionism tries to explain the long persistence of this dogma by two facts: firstly the obvious non-contradictority of the principle for an arbitrary single assertion; secondly the practical validity of the whole of classical logic for an extensive group of simple every day phenomena. The latter fact apparently made such a strong impression that the play of thought that classical logic originally was, became a deeprooted habit of thought which was considered not only as useful but even as aprioristic.

Obviously the field of validity of the principle of the excluded third is identical with the intersection of the fields of validity of the principle of testability and the principle of reciprocity of complementarity. Furthermore the former field of validity is a proper subfield of each of the latter ones, as is shown by the following examples:

Let A be the species of the direct checking-numbers of drifts with rational counting-numbers, B the species of the irrational real numbers, C the union of A and B. Then all assertions of rationality of an element of C satisfy the principle of testability, whilst there are assertions of rationality of an element of C not satisfying the principle of the excluded third. Again, all assertions of equality of two real numbers satisfy the principle of reciprocity of complementarity, whereas there are assertions of equality of two real numbers not satisfying the principle of the excluded third.

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In the domain of mathematical assertions the property of absurdity, just as the property of truth, is a universally additive property, that is to say, if it holds for each element a of a species of assertions, it also holds for the assertion which is the union of the assertions a. This property of universal additivity does not obtain for the property of non-contradictority. However, non-contradictority does possess the weaker property of finite additivity, that is to say, if the assertions e and a are non-contradictory, the assertion -r which is the union of e and a, is also non-contradictory. For, let us start for a moment from the supposition w that -r is contradictory. Then the truth of e would entail the contradictority of a, which would clash with the data, so that the truth of e is absurd, i.e. e is absurd. This consequence of the supposition w clashing with the data, the supposition w is contradictory, i.e. -r is non-contradictory.

Application of this theorem to the special non-contradictory assertions that are the enunciations of the principle of the excluded third for a single assertion, establishes the above-mentioned non-contradictority of the simultaneous enunciation of this principle for a finite number of assertions.

Within some species of mathematical entities the absurdities of two non-equivalent 1) assertions may be equivalent. E.g. each of the following three pairs of non-equivalent assertions relative to a real number a:

I 1 . a = a; II 1 . a > O;

III 1 . a > O;

I 2. either a :S 0 or a > 0 II 2. either a = 0 or a 0> 0

III 2. a o> 0

furnishes a pair of equivalent absurdities.

It occurs that within some species of mathematical entities some absurdities of constructive properties can be given a constructive form. E.g. for a natural number a the absurdity of the existence of two natural numbers different from a and from 1 and having a as their product, is equivalent to the existence, whenever a is divided by a natural number different from a and from 1 , of a remainder. Likewise, for two real numbers a and b the relation a > b introduced above as an absurdity of a constructive property can be formulated constructively as follows: Let ai, a2, . . . and bi, b2, . . . be convergent infinite sequences of rational numbers ddining a and b respectively. Then, for any natural number n, a natural number m can be calculated such that av - bv 0> - 2-n for v > m.

On the other hand there seems to be little hope for r·educing irrationality of a real number a, or one of the relations a -=/:. b and a > b for real numbers a and b, to a constructive property, if we remark that a direct checking-

1) By non-equivalence we understand absurdity of equivalence, just as by noncontradictority we understand absurdity of contradictority.

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number of a drift whose kernel is rational and whose counting-numbers are irrational, is irrational without lying apart from the species of rational numbers; further that a direct checking-number of an arbitrary drift differs from the kernel of the drift without lying apart from it, and that a .direct checking-number of a right-winged drift lies to the right of the

[5] kernel of the drift without lying apart from it.

It occurs that within some species of mathematical entltles some noncontradictorities of constructive properties C can be given either a constructive form (possibly, but not necessarily, in consequence of reciprocity of complementarity holding for C) or the form of an absurdity of a constructive property. E.g. for real numbers a and b the non-contradictority of a = b is equivalent to a = b, and the non-contradictority of: either a = b or a 0> b, is equivalent to a > b; further the non-contradictority of a 0> b is equivalent to the absur·dity of a ::=:: b as well as to the absurdity of: either a = b or a <0 b.

On the other hand, if we think of the property of non-contradictority of rationality existing for all direct checking-numbers of drifts whose counting-numbers are rational, there seems to be little hope for reducing non-contradictority of rationality of a real number to a constructive property or to an absurdity of a constructive property.

If we understand by the simple absurdity of the property r; the absurdity of r;, and by the (n + 1 )-fold absurdity of r; the absurdity of the n-fold absurdity of r;, than a theorem established above expresses that threefold absurdity is equivalent to simple absurdity. And a corollary of this theorem is that n-fold absurdity is equivalent to simple or to double absurdity

[6] according as n is odd or even.

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I should like to terminate here. I hope I have made clear that intuitionism on the one hand subtilizes logic, on the other hand denounces logic as a source of truth. Further that intuitionistic mathematics is inner architecture, and that research in foundations of mathematics is inner inquiry with revealing and liberating consequences, also in non-mathematical domains of thought.

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